How to get file names from the directory, not the entire path

Try GetFileName() method:

Path.GetFileName(filename);

System.IO.Path is your friend here:

var filenames = from fullFilename
                in Directory.EnumerateFiles(targetdirectory,"backup-*.zip")
                select Path.GetFileName(fullFilename);

foreach (string filename in filenames)
{
    // ...
}

You could use the GetFileName method to extract only the filename without a path:

string filenameWithoutPath = Path.GetFileName(filename);